In-class Exercise 5

Published

December 17, 2022

Lesson 5: Geographically Weighted Logistic Regression Model

1 Overview

To build an explanatory model to discover factor affecting water point status in Osun State, Nigeria

1.1 Study area

Osun State, Nigeria

1.2 Data sets

Osun.rds, Osun_wp_sf.rds

1.3 Model Variables

  • Dependent variable: water point status

  • Independent variables:

  • distance_to_primary_road,

  • distance_to_secondary_road,

  • distance_to_city,

  • distance_to_town,

  • water_point_population,

  • local_population_1km,

  • usage_capacity,

  • is_urban,

  • water_source_clean

2 Getting Started

Load the following R packages:

pacman::p_load(sf, tidyverse, funModeling, 
               blorr, corrplot, ggpubr, 
               spdep, GWmodel, tmap, skimr, caret)

3 Data Import

3.1 Importing water point data

wp_nga <- read_csv("aspatial/WPdx.csv") %>%
  filter(`#clean_country_name` == "Nigeria")

3.2 Importing the analytical data

Osun <- read_rds("rds/Osun.rds")
Osun_wp_sf <- read_rds("rds/Osun_wp_sf.rds")
Osun_wp_sf %>%
  freq(input = 'status')

  status frequency percentage cumulative_perc
1   TRUE      2642       55.5            55.5
2  FALSE      2118       44.5           100.0
tmap_mode("view")
tm_shape(Osun)+
 tmap_options(check.and.fix = TRUE) +
  tm_polygons(alpha=0.4)+
tm_shape(Osun_wp_sf)+
  tm_dots(col = "status",
          alpha=0.6)+
  tm_view(set.zoom.limits = c(9,12))

4 EDA

Summary Statistics with skimr:

Osun_wp_sf %>%
  skim()
Data summary
Name Piped data
Number of rows 4760
Number of columns 75
_______________________
Column type frequency:
character 47
logical 5
numeric 23
________________________
Group variables None

Variable type: character

skim_variable n_missing complete_rate min max empty n_unique whitespace
source 0 1.00 5 44 0 2 0
report_date 0 1.00 22 22 0 42 0
status_id 0 1.00 2 7 0 3 0
water_source_clean 0 1.00 8 22 0 3 0
water_source_category 0 1.00 4 6 0 2 0
water_tech_clean 24 0.99 9 23 0 3 0
water_tech_category 24 0.99 9 15 0 2 0
facility_type 0 1.00 8 8 0 1 0
clean_country_name 0 1.00 7 7 0 1 0
clean_adm1 0 1.00 3 5 0 5 0
clean_adm2 0 1.00 3 14 0 35 0
clean_adm3 4760 0.00 NA NA 0 0 0
clean_adm4 4760 0.00 NA NA 0 0 0
installer 4760 0.00 NA NA 0 0 0
management_clean 1573 0.67 5 37 0 7 0
status_clean 0 1.00 9 32 0 7 0
pay 0 1.00 2 39 0 7 0
fecal_coliform_presence 4760 0.00 NA NA 0 0 0
subjective_quality 0 1.00 18 20 0 4 0
activity_id 4757 0.00 36 36 0 3 0
scheme_id 4760 0.00 NA NA 0 0 0
wpdx_id 0 1.00 12 12 0 4760 0
notes 0 1.00 2 96 0 3502 0
orig_lnk 4757 0.00 84 84 0 1 0
photo_lnk 41 0.99 84 84 0 4719 0
country_id 0 1.00 2 2 0 1 0
data_lnk 0 1.00 79 96 0 2 0
water_point_history 0 1.00 142 834 0 4750 0
clean_country_id 0 1.00 3 3 0 1 0
country_name 0 1.00 7 7 0 1 0
water_source 0 1.00 8 30 0 4 0
water_tech 0 1.00 5 37 0 20 0
adm2 0 1.00 3 14 0 33 0
adm3 4760 0.00 NA NA 0 0 0
management 1573 0.67 5 47 0 7 0
adm1 0 1.00 4 5 0 4 0
New Georeferenced Column 0 1.00 16 35 0 4760 0
lat_lon_deg 0 1.00 13 32 0 4760 0
public_data_source 0 1.00 84 102 0 2 0
converted 0 1.00 53 53 0 1 0
created_timestamp 0 1.00 22 22 0 2 0
updated_timestamp 0 1.00 22 22 0 2 0
Geometry 0 1.00 33 37 0 4760 0
ADM2_EN 0 1.00 3 14 0 30 0
ADM2_PCODE 0 1.00 8 8 0 30 0
ADM1_EN 0 1.00 4 4 0 1 0
ADM1_PCODE 0 1.00 5 5 0 1 0

Variable type: logical

skim_variable n_missing complete_rate mean count
rehab_year 4760 0 NaN :
rehabilitator 4760 0 NaN :
is_urban 0 1 0.39 FAL: 2884, TRU: 1876
latest_record 0 1 1.00 TRU: 4760
status 0 1 0.56 TRU: 2642, FAL: 2118

Variable type: numeric

skim_variable n_missing complete_rate mean sd p0 p25 p50 p75 p100 hist
row_id 0 1.00 68550.48 10216.94 49601.00 66874.75 68244.50 69562.25 471319.00 ▇▁▁▁▁
lat_deg 0 1.00 7.68 0.22 7.06 7.51 7.71 7.88 8.06 ▁▂▇▇▇
lon_deg 0 1.00 4.54 0.21 4.08 4.36 4.56 4.71 5.06 ▃▆▇▇▂
install_year 1144 0.76 2008.63 6.04 1917.00 2006.00 2010.00 2013.00 2015.00 ▁▁▁▁▇
fecal_coliform_value 4760 0.00 NaN NA NA NA NA NA NA
distance_to_primary_road 0 1.00 5021.53 5648.34 0.01 719.36 2972.78 7314.73 26909.86 ▇▂▁▁▁
distance_to_secondary_road 0 1.00 3750.47 3938.63 0.15 460.90 2554.25 5791.94 19559.48 ▇▃▁▁▁
distance_to_tertiary_road 0 1.00 1259.28 1680.04 0.02 121.25 521.77 1834.42 10966.27 ▇▂▁▁▁
distance_to_city 0 1.00 16663.99 10960.82 53.05 7930.75 15030.41 24255.75 47934.34 ▇▇▆▃▁
distance_to_town 0 1.00 16726.59 12452.65 30.00 6876.92 12204.53 27739.46 44020.64 ▇▅▃▃▂
rehab_priority 2654 0.44 489.33 1658.81 0.00 7.00 91.50 376.25 29697.00 ▇▁▁▁▁
water_point_population 4 1.00 513.58 1458.92 0.00 14.00 119.00 433.25 29697.00 ▇▁▁▁▁
local_population_1km 4 1.00 2727.16 4189.46 0.00 176.00 1032.00 3717.00 36118.00 ▇▁▁▁▁
crucialness_score 798 0.83 0.26 0.28 0.00 0.07 0.15 0.35 1.00 ▇▃▁▁▁
pressure_score 798 0.83 1.46 4.16 0.00 0.12 0.41 1.24 93.69 ▇▁▁▁▁
usage_capacity 0 1.00 560.74 338.46 300.00 300.00 300.00 1000.00 1000.00 ▇▁▁▁▅
days_since_report 0 1.00 2692.69 41.92 1483.00 2688.00 2693.00 2700.00 4645.00 ▁▇▁▁▁
staleness_score 0 1.00 42.80 0.58 23.13 42.70 42.79 42.86 62.66 ▁▁▇▁▁
location_id 0 1.00 235865.49 6657.60 23741.00 230638.75 236199.50 240061.25 267454.00 ▁▁▁▁▇
cluster_size 0 1.00 1.05 0.25 1.00 1.00 1.00 1.00 4.00 ▇▁▁▁▁
lat_deg_original 4760 0.00 NaN NA NA NA NA NA NA
lon_deg_original 4760 0.00 NaN NA NA NA NA NA NA
count 0 1.00 1.00 0.00 1.00 1.00 1.00 1.00 1.00 ▁▁▇▁▁
Osun_wp_sf_clean <- Osun_wp_sf %>%
  filter_at(vars(status,
                 distance_to_primary_road,
                 distance_to_secondary_road,
                 distance_to_tertiary_road,
                 distance_to_city,
                 distance_to_town,
                 water_point_population,
                 local_population_1km,
                 usage_capacity,
                 is_urban,
                 water_source_clean),
            all_vars(!is.na(.))) %>%
  mutate(usage_capacity = as.factor(usage_capacity))

5 Correlation Analysis

Osun_wp <- Osun_wp_sf_clean %>%
  select(c(7,35:39,42:43,46:47,57)) %>%
  st_set_geometry(NULL)
cluster_vars.cor = cor(
  Osun_wp[,2:7])
corrplot.mixed(cluster_vars.cor,
               lower = "ellipse",
               upper = "number",
               tl.pos = "lt",
               diag = "l",
               tl.col = "black")

6 Building a Logistic Regression

model <- glm(status ~ distance_to_primary_road +
               distance_to_secondary_road +
               distance_to_tertiary_road +
               distance_to_city +
               distance_to_town +
               is_urban +
               usage_capacity +
               water_source_clean +
               water_point_population +
               local_population_1km,
             data = Osun_wp_sf_clean,
             family = binomial(link = 'logit'))

Instead of using typical R report, blr_regress() of blorr package is used.

blr_regress(model)
                             Model Overview                              
------------------------------------------------------------------------
Data Set    Resp Var    Obs.    Df. Model    Df. Residual    Convergence 
------------------------------------------------------------------------
  data       status     4756      4755           4744           TRUE     
------------------------------------------------------------------------

                    Response Summary                     
--------------------------------------------------------
Outcome        Frequency        Outcome        Frequency 
--------------------------------------------------------
   0             2114              1             2642    
--------------------------------------------------------

                                 Maximum Likelihood Estimates                                   
-----------------------------------------------------------------------------------------------
               Parameter                    DF    Estimate    Std. Error    z value     Pr(>|z|) 
-----------------------------------------------------------------------------------------------
              (Intercept)                   1      0.3887        0.1124      3.4588       5e-04 
        distance_to_primary_road            1      0.0000        0.0000     -0.7153      0.4744 
       distance_to_secondary_road           1      0.0000        0.0000     -0.5530      0.5802 
       distance_to_tertiary_road            1      1e-04         0.0000      4.6708      0.0000 
            distance_to_city                1      0.0000        0.0000     -4.7574      0.0000 
            distance_to_town                1      0.0000        0.0000     -4.9170      0.0000 
              is_urbanTRUE                  1     -0.2971        0.0819     -3.6294       3e-04 
           usage_capacity1000               1     -0.6230        0.0697     -8.9366      0.0000 
water_source_cleanProtected Shallow Well    1      0.5040        0.0857      5.8783      0.0000 
   water_source_cleanProtected Spring       1      1.2882        0.4388      2.9359      0.0033 
         water_point_population             1      -5e-04        0.0000    -11.3686      0.0000 
          local_population_1km              1      3e-04         0.0000     19.2953      0.0000 
-----------------------------------------------------------------------------------------------

 Association of Predicted Probabilities and Observed Responses  
---------------------------------------------------------------
% Concordant          0.7347          Somers' D        0.4693   
% Discordant          0.2653          Gamma            0.4693   
% Tied                0.0000          Tau-a            0.2318   
Pairs                5585188          c                0.7347   
---------------------------------------------------------------

In the code chunk below, blr_confusion_matrix() of blorr package is used to compute the confusion matrix of the estimated outcomes by using 0.5 as the cutoff value.

blr_confusion_matrix(model, cutoff = 0.5)
Confusion Matrix and Statistics 

          Reference
Prediction FALSE TRUE
         0  1301  738
         1   813 1904

                Accuracy : 0.6739 
     No Information Rate : 0.4445 

                   Kappa : 0.3373 

McNemars's Test P-Value  : 0.0602 

             Sensitivity : 0.7207 
             Specificity : 0.6154 
          Pos Pred Value : 0.7008 
          Neg Pred Value : 0.6381 
              Prevalence : 0.5555 
          Detection Rate : 0.4003 
    Detection Prevalence : 0.5713 
       Balanced Accuracy : 0.6680 
               Precision : 0.7008 
                  Recall : 0.7207 

        'Positive' Class : 1

New model:

model_new <- glm(status ~ distance_to_tertiary_road +
               distance_to_city +
               distance_to_town +
               is_urban +
               usage_capacity +
               water_source_clean +
               water_point_population +
               local_population_1km,
             data = Osun_wp_sf_clean,
             family = binomial(link = 'logit'))
blr_regress(model_new)
                             Model Overview                              
------------------------------------------------------------------------
Data Set    Resp Var    Obs.    Df. Model    Df. Residual    Convergence 
------------------------------------------------------------------------
  data       status     4756      4755           4746           TRUE     
------------------------------------------------------------------------

                    Response Summary                     
--------------------------------------------------------
Outcome        Frequency        Outcome        Frequency 
--------------------------------------------------------
   0             2114              1             2642    
--------------------------------------------------------

                                 Maximum Likelihood Estimates                                   
-----------------------------------------------------------------------------------------------
               Parameter                    DF    Estimate    Std. Error    z value     Pr(>|z|) 
-----------------------------------------------------------------------------------------------
              (Intercept)                   1      0.3540        0.1055      3.3541       8e-04 
       distance_to_tertiary_road            1      1e-04         0.0000      4.9096      0.0000 
            distance_to_city                1      0.0000        0.0000     -5.2022      0.0000 
            distance_to_town                1      0.0000        0.0000     -5.4660      0.0000 
              is_urbanTRUE                  1     -0.2667        0.0747     -3.5690       4e-04 
           usage_capacity1000               1     -0.6206        0.0697     -8.9081      0.0000 
water_source_cleanProtected Shallow Well    1      0.4947        0.0850      5.8228      0.0000 
   water_source_cleanProtected Spring       1      1.2790        0.4384      2.9174      0.0035 
         water_point_population             1      -5e-04        0.0000    -11.3902      0.0000 
          local_population_1km              1      3e-04         0.0000     19.4069      0.0000 
-----------------------------------------------------------------------------------------------

 Association of Predicted Probabilities and Observed Responses  
---------------------------------------------------------------
% Concordant          0.7349          Somers' D        0.4697   
% Discordant          0.2651          Gamma            0.4697   
% Tied                0.0000          Tau-a            0.2320   
Pairs                5585188          c                0.7349   
---------------------------------------------------------------
blr_confusion_matrix(model_new, cutoff = 0.5)
Confusion Matrix and Statistics 

          Reference
Prediction FALSE TRUE
         0  1300  743
         1   814 1899

                Accuracy : 0.6726 
     No Information Rate : 0.4445 

                   Kappa : 0.3348 

McNemars's Test P-Value  : 0.0761 

             Sensitivity : 0.7188 
             Specificity : 0.6149 
          Pos Pred Value : 0.7000 
          Neg Pred Value : 0.6363 
              Prevalence : 0.5555 
          Detection Rate : 0.3993 
    Detection Prevalence : 0.5704 
       Balanced Accuracy : 0.6669 
               Precision : 0.7000 
                  Recall : 0.7188 

        'Positive' Class : 1

7 Building Geographically Weight

7.1 Converting from sf to sp data frame

Osun_wp_sp <- Osun_wp_sf_clean %>%
  select(c(status,
           distance_to_primary_road,
           distance_to_secondary_road,
           distance_to_tertiary_road,
           distance_to_city,
           distance_to_town,
           water_point_population,
           local_population_1km,
           usage_capacity,
           is_urban,
           water_source_clean)) %>%
  as_Spatial()
Osun_wp_sp
class       : SpatialPointsDataFrame 
features    : 4756 
extent      : 182502.4, 290751, 340054.1, 450905.3  (xmin, xmax, ymin, ymax)
crs         : +proj=tmerc +lat_0=4 +lon_0=8.5 +k=0.99975 +x_0=670553.98 +y_0=0 +a=6378249.145 +rf=293.465 +towgs84=-92,-93,122,0,0,0,0 +units=m +no_defs 
variables   : 11
names       : status, distance_to_primary_road, distance_to_secondary_road, distance_to_tertiary_road, distance_to_city, distance_to_town, water_point_population, local_population_1km, usage_capacity, is_urban, water_source_clean 
min values  :      0,        0.014461356813335,          0.152195902540837,         0.017815121653488, 53.0461399623541, 30.0019777713073,                      0,                    0,           1000,        0,           Borehole 
max values  :      1,         26909.8616132094,           19559.4793799085,          10966.2705628969,  47934.343603562, 44020.6393368124,                  29697,                36118,            300,        1,   Protected Spring 

7.2 Building Fixed Bandwidth GWR Model

Computing fixed bandwidth:

bw.fixed <- bw.ggwr(status ~ 
                      distance_to_primary_road +
                       distance_to_secondary_road +
                       distance_to_tertiary_road +
                       distance_to_city +
                       distance_to_town +
                       is_urban +
                       usage_capacity +
                       water_source_clean +
                       water_point_population +
                       local_population_1km,
                    data = Osun_wp_sp,
                    family = "binomial",
                    approach = "AIC",
                    kernel = "gaussian",
                    adaptive = FALSE,
                    longlat = FALSE)
bw.fixed
gwlr.fixed <- ggwr.basic(status ~ 
                      distance_to_primary_road +
                       distance_to_secondary_road +
                       distance_to_tertiary_road +
                       distance_to_city +
                       distance_to_town +
                       is_urban +
                       usage_capacity +
                       water_source_clean +
                       water_point_population +
                       local_population_1km,
                      data = Osun_wp_sp,
                      bw = 2599.672,
                      family = "binomial",
                      kernel = "gaussian",
                      adaptive = FALSE,
                      longlat = FALSE)
 Iteration    Log-Likelihood
=========================
       0        -1958 
       1        -1676 
       2        -1526 
       3        -1443 
       4        -1405 
       5        -1405 
gwlr.fixed
   ***********************************************************************
   *                       Package   GWmodel                             *
   ***********************************************************************
   Program starts at: 2022-12-17 15:17:59 
   Call:
   ggwr.basic(formula = status ~ distance_to_primary_road + distance_to_secondary_road + 
    distance_to_tertiary_road + distance_to_city + distance_to_town + 
    is_urban + usage_capacity + water_source_clean + water_point_population + 
    local_population_1km, data = Osun_wp_sp, bw = 2599.672, family = "binomial", 
    kernel = "gaussian", adaptive = FALSE, longlat = FALSE)

   Dependent (y) variable:  status
   Independent variables:  distance_to_primary_road distance_to_secondary_road distance_to_tertiary_road distance_to_city distance_to_town is_urban usage_capacity water_source_clean water_point_population local_population_1km
   Number of data points: 4756
   Used family: binomial
   ***********************************************************************
   *              Results of Generalized linear Regression               *
   ***********************************************************************

Call:
NULL

Deviance Residuals: 
     Min        1Q    Median        3Q       Max  
-124.555    -1.755     1.072     1.742    34.333  

Coefficients:
                                           Estimate Std. Error z value Pr(>|z|)
Intercept                                 3.887e-01  1.124e-01   3.459 0.000543
distance_to_primary_road                 -4.642e-06  6.490e-06  -0.715 0.474422
distance_to_secondary_road               -5.143e-06  9.299e-06  -0.553 0.580230
distance_to_tertiary_road                 9.683e-05  2.073e-05   4.671 3.00e-06
distance_to_city                         -1.686e-05  3.544e-06  -4.757 1.96e-06
distance_to_town                         -1.480e-05  3.009e-06  -4.917 8.79e-07
is_urbanTRUE                             -2.971e-01  8.185e-02  -3.629 0.000284
usage_capacity1000                       -6.230e-01  6.972e-02  -8.937  < 2e-16
water_source_cleanProtected Shallow Well  5.040e-01  8.574e-02   5.878 4.14e-09
water_source_cleanProtected Spring        1.288e+00  4.388e-01   2.936 0.003325
water_point_population                   -5.097e-04  4.484e-05 -11.369  < 2e-16
local_population_1km                      3.451e-04  1.788e-05  19.295  < 2e-16
                                            
Intercept                                ***
distance_to_primary_road                    
distance_to_secondary_road                  
distance_to_tertiary_road                ***
distance_to_city                         ***
distance_to_town                         ***
is_urbanTRUE                             ***
usage_capacity1000                       ***
water_source_cleanProtected Shallow Well ***
water_source_cleanProtected Spring       ** 
water_point_population                   ***
local_population_1km                     ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

(Dispersion parameter for binomial family taken to be 1)

    Null deviance: 6534.5  on 4755  degrees of freedom
Residual deviance: 5688.0  on 4744  degrees of freedom
AIC: 5712

Number of Fisher Scoring iterations: 5


 AICc:  5712.099
 Pseudo R-square value:  0.1295351
   ***********************************************************************
   *          Results of Geographically Weighted Regression              *
   ***********************************************************************

   *********************Model calibration information*********************
   Kernel function: gaussian 
   Fixed bandwidth: 2599.672 
   Regression points: the same locations as observations are used.
   Distance metric: A distance matrix is specified for this model calibration.

   ************Summary of Generalized GWR coefficient estimates:**********
                                                   Min.     1st Qu.      Median
   Intercept                                -8.7229e+02 -4.9955e+00  1.7600e+00
   distance_to_primary_road                 -1.9389e-02 -4.8031e-04  2.9618e-05
   distance_to_secondary_road               -1.5921e-02 -3.7551e-04  1.2317e-04
   distance_to_tertiary_road                -1.5618e-02 -4.2368e-04  7.6179e-05
   distance_to_city                         -1.8416e-02 -5.6217e-04 -1.2726e-04
   distance_to_town                         -2.2411e-02 -5.7283e-04 -1.5155e-04
   is_urbanTRUE                             -1.9790e+02 -4.2908e+00 -1.6864e+00
   usage_capacity1000                       -2.0772e+01 -9.7231e-01 -4.1592e-01
   water_source_cleanProtected.Shallow.Well -2.0789e+01 -4.5190e-01  5.3340e-01
   water_source_cleanProtected.Spring       -5.2235e+02 -5.5977e+00  2.5441e+00
   water_point_population                   -5.2208e-02 -2.2767e-03 -9.8875e-04
   local_population_1km                     -1.2698e-01  4.9952e-04  1.0638e-03
                                                3rd Qu.      Max.
   Intercept                                 1.2763e+01 1073.2156
   distance_to_primary_road                  4.8443e-04    0.0142
   distance_to_secondary_road                6.0692e-04    0.0258
   distance_to_tertiary_road                 6.6815e-04    0.0128
   distance_to_city                          2.3718e-04    0.0150
   distance_to_town                          1.9271e-04    0.0224
   is_urbanTRUE                              1.2841e+00  744.3099
   usage_capacity1000                        3.0322e-01    5.9281
   water_source_cleanProtected.Shallow.Well  1.7849e+00   67.6343
   water_source_cleanProtected.Spring        6.7663e+00  317.4133
   water_point_population                    5.0102e-04    0.1309
   local_population_1km                      1.8157e-03    0.0392
   ************************Diagnostic information*************************
   Number of data points: 4756 
   GW Deviance: 2795.084 
   AIC : 4414.606 
   AICc : 4747.423 
   Pseudo R-square value:  0.5722559 

   ***********************************************************************
   Program stops at: 2022-12-17 15:18:56 

New model:

bw.fixed_new <- bw.ggwr(status ~ 
                       distance_to_tertiary_road +
                       distance_to_city +
                       distance_to_town +
                       is_urban +
                       usage_capacity +
                       water_source_clean +
                       water_point_population +
                       local_population_1km,
                    data = Osun_wp_sp,
                    family = "binomial",
                    approach = "AIC",
                    kernel = "gaussian",
                    adaptive = FALSE,
                    longlat = FALSE)
bw.fixed_new
gwlr.fixed_new <- ggwr.basic(status ~ 
                       distance_to_tertiary_road +
                       distance_to_city +
                       distance_to_town +
                       is_urban +
                       usage_capacity +
                       water_source_clean +
                       water_point_population +
                       local_population_1km,
                      data = Osun_wp_sp,
                      bw = 2377.371,
                      family = "binomial",
                      kernel = "gaussian",
                      adaptive = FALSE,
                      longlat = FALSE)
 Iteration    Log-Likelihood
=========================
       0        -1959 
       1        -1680 
       2        -1531 
       3        -1447 
       4        -1413 
       5        -1413 
gwlr.fixed_new
   ***********************************************************************
   *                       Package   GWmodel                             *
   ***********************************************************************
   Program starts at: 2022-12-17 15:18:56 
   Call:
   ggwr.basic(formula = status ~ distance_to_tertiary_road + distance_to_city + 
    distance_to_town + is_urban + usage_capacity + water_source_clean + 
    water_point_population + local_population_1km, data = Osun_wp_sp, 
    bw = 2377.371, family = "binomial", kernel = "gaussian", 
    adaptive = FALSE, longlat = FALSE)

   Dependent (y) variable:  status
   Independent variables:  distance_to_tertiary_road distance_to_city distance_to_town is_urban usage_capacity water_source_clean water_point_population local_population_1km
   Number of data points: 4756
   Used family: binomial
   ***********************************************************************
   *              Results of Generalized linear Regression               *
   ***********************************************************************

Call:
NULL

Deviance Residuals: 
     Min        1Q    Median        3Q       Max  
-129.368    -1.750     1.074     1.742    34.126  

Coefficients:
                                           Estimate Std. Error z value Pr(>|z|)
Intercept                                 3.540e-01  1.055e-01   3.354 0.000796
distance_to_tertiary_road                 1.001e-04  2.040e-05   4.910 9.13e-07
distance_to_city                         -1.764e-05  3.391e-06  -5.202 1.97e-07
distance_to_town                         -1.544e-05  2.825e-06  -5.466 4.60e-08
is_urbanTRUE                             -2.667e-01  7.474e-02  -3.569 0.000358
usage_capacity1000                       -6.206e-01  6.966e-02  -8.908  < 2e-16
water_source_cleanProtected Shallow Well  4.947e-01  8.496e-02   5.823 5.79e-09
water_source_cleanProtected Spring        1.279e+00  4.384e-01   2.917 0.003530
water_point_population                   -5.098e-04  4.476e-05 -11.390  < 2e-16
local_population_1km                      3.452e-04  1.779e-05  19.407  < 2e-16
                                            
Intercept                                ***
distance_to_tertiary_road                ***
distance_to_city                         ***
distance_to_town                         ***
is_urbanTRUE                             ***
usage_capacity1000                       ***
water_source_cleanProtected Shallow Well ***
water_source_cleanProtected Spring       ** 
water_point_population                   ***
local_population_1km                     ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

(Dispersion parameter for binomial family taken to be 1)

    Null deviance: 6534.5  on 4755  degrees of freedom
Residual deviance: 5688.9  on 4746  degrees of freedom
AIC: 5708.9

Number of Fisher Scoring iterations: 5


 AICc:  5708.923
 Pseudo R-square value:  0.129406
   ***********************************************************************
   *          Results of Geographically Weighted Regression              *
   ***********************************************************************

   *********************Model calibration information*********************
   Kernel function: gaussian 
   Fixed bandwidth: 2377.371 
   Regression points: the same locations as observations are used.
   Distance metric: A distance matrix is specified for this model calibration.

   ************Summary of Generalized GWR coefficient estimates:**********
                                                   Min.     1st Qu.      Median
   Intercept                                -3.7021e+02 -4.3797e+00  3.5590e+00
   distance_to_tertiary_road                -3.1622e-02 -4.5462e-04  9.1291e-05
   distance_to_city                         -5.4555e-02 -6.5623e-04 -1.3507e-04
   distance_to_town                         -8.6549e-03 -5.2754e-04 -1.6785e-04
   is_urbanTRUE                             -7.3554e+02 -3.4675e+00 -1.6596e+00
   usage_capacity1000                       -5.5889e+01 -1.0347e+00 -4.1960e-01
   water_source_cleanProtected.Shallow.Well -1.8842e+02 -4.7295e-01  6.2378e-01
   water_source_cleanProtected.Spring       -1.3630e+03 -5.3436e+00  2.7714e+00
   water_point_population                   -2.9696e-02 -2.2705e-03 -1.2277e-03
   local_population_1km                     -7.7730e-02  4.4281e-04  1.0548e-03
                                                3rd Qu.      Max.
   Intercept                                 1.3755e+01 2171.6375
   distance_to_tertiary_road                 6.3011e-04    0.0237
   distance_to_city                          1.5921e-04    0.0162
   distance_to_town                          2.4490e-04    0.0179
   is_urbanTRUE                              1.0554e+00  995.1841
   usage_capacity1000                        3.9113e-01    9.2449
   water_source_cleanProtected.Shallow.Well  1.9564e+00   66.8914
   water_source_cleanProtected.Spring        7.0805e+00  208.3749
   water_point_population                    4.5879e-04    0.0765
   local_population_1km                      1.8479e-03    0.0333
   ************************Diagnostic information*************************
   Number of data points: 4756 
   GW Deviance: 2815.659 
   AIC : 4418.776 
   AICc : 4744.213 
   Pseudo R-square value:  0.5691072 

   ***********************************************************************
   Program stops at: 2022-12-17 15:19:45 

7.3 Model Assessment

7.3.1 Converting SDF into sf data.frame

To assess the performance of the gwLR, firstly, we will convert the SDF object in as data frame by using the code chunk below.

gwr.fixed <- as.data.frame(gwlr.fixed$SDF)

New model:

gwr.fixed_new <- as.data.frame(gwlr.fixed_new$SDF)

Next, we will label yhat values greater or equal to 0.5 into 1 and else 0. The result of the logic comparison operation will be saved into a field called most.

gwr.fixed <- gwr.fixed %>%
  mutate(most = ifelse(
    gwr.fixed$yhat >= 0.5,T,F))
gwr.fixed$y <- as.factor(gwr.fixed$y)
gwr.fixed$most <- as.factor(gwr.fixed$most)
CM <- confusionMatrix(data=gwr.fixed$most, reference=gwr.fixed$y)
CM
Confusion Matrix and Statistics

          Reference
Prediction FALSE TRUE
     FALSE  1824  263
     TRUE    290 2379
                                          
               Accuracy : 0.8837          
                 95% CI : (0.8743, 0.8927)
    No Information Rate : 0.5555          
    P-Value [Acc > NIR] : <2e-16          
                                          
                  Kappa : 0.7642          
                                          
 Mcnemar's Test P-Value : 0.2689          
                                          
            Sensitivity : 0.8628          
            Specificity : 0.9005          
         Pos Pred Value : 0.8740          
         Neg Pred Value : 0.8913          
             Prevalence : 0.4445          
         Detection Rate : 0.3835          
   Detection Prevalence : 0.4388          
      Balanced Accuracy : 0.8816          
                                          
       'Positive' Class : FALSE           
                                          
Osun_wp_sf_selected <- Osun_wp_sf_clean %>%
  select(c(ADM2_EN, ADM2_PCODE,
           ADM1_EN, ADM1_PCODE,
           status))
gwr_sf.fixed <- cbind(Osun_wp_sf_selected,gwr.fixed)

New model:

gwr.fixed_new <- gwr.fixed_new %>%
  mutate(most = ifelse(
    gwr.fixed_new$yhat >= 0.5,T,F))

gwr.fixed_new$y <- as.factor(gwr.fixed_new$y)
gwr.fixed_new$most <- as.factor(gwr.fixed_new$most)
CM_new <- confusionMatrix(data=gwr.fixed_new$most, reference=gwr.fixed_new$y)
CM_new
Confusion Matrix and Statistics

          Reference
Prediction FALSE TRUE
     FALSE  1833  268
     TRUE    281 2374
                                          
               Accuracy : 0.8846          
                 95% CI : (0.8751, 0.8935)
    No Information Rate : 0.5555          
    P-Value [Acc > NIR] : <2e-16          
                                          
                  Kappa : 0.7661          
                                          
 Mcnemar's Test P-Value : 0.6085          
                                          
            Sensitivity : 0.8671          
            Specificity : 0.8986          
         Pos Pred Value : 0.8724          
         Neg Pred Value : 0.8942          
             Prevalence : 0.4445          
         Detection Rate : 0.3854          
   Detection Prevalence : 0.4418          
      Balanced Accuracy : 0.8828          
                                          
       'Positive' Class : FALSE           
                                          
gwr_sf.fixed_new <- cbind(Osun_wp_sf_selected,gwr.fixed_new)

7.4 Visualising gwLR

Visualising coefficient estimates:

tmap_mode("view")
prob_T <- tm_shape(Osun) +
  tm_polygons(alpha = 0.1) +
tm_shape(gwr_sf.fixed) + 
  tm_dots(col = "yhat",
          border.col = "gray60",
          border.lwd = 1) +
  tm_view(set.zoom.limits = c(8,14))
prob_T
tertiary_TV <- tm_shape(Osun) +
  tm_polygons(alpha = 0.1) +
tm_shape(gwr_sf.fixed) +
  tm_dots(col = "distance_to_tertiary_road_TV",
          border.col = "gray60",
          border.lwd = 1) +
  tm_view(set.zoom.limits = c(8,14))

tertiary_SE <- tm_shape(Osun) +
  tm_polygons(alpha = 0.1) +
tm_shape(gwr_sf.fixed) +
  tm_dots(col = "distance_to_tertiary_road_SE",
          border.col = "gray60",
          border.lwd = 1) +
  tm_view(set.zoom.limits = c(8,14))

tmap_arrange(tertiary_SE,tertiary_TV,asp = 1,ncol = 2,sync = TRUE)